Everything you're saying here is true, but I think you're forgetting something incredibly important that isn't easily representable on a Turing machine at all:

Everything you're saying here is true, but I think you're forgetting something incredibly important that isn't easily representable on a Turing machine at all: equivalences of the terms and their rewrite rules. Lambda calculus provides you with a set of ways in which you could rewrite your program and get an equivalent one (by the definition of lambda calculus), or how to execute the program itself. It means that most of the functionality for deciding when the code should be executed (by the compiler or at run time) is already in the compiler. It's something that imperative languages struggle with.